Skip to main content
SpringerLink
Log in

Estimates for Order Statistics in Terms of Quantiles

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Let X1, . . .,Xn be independent nonnegative random variables with cumulative distribution functions F1, F2, . . . , Fn satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X1, . . . , Xn) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution \( F=\frac{1}{n}\sum \limits_{i=1}^n{F}_i \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. C. Arnold and N. Balakrishnan, Relations, Bounds and Approximations for Order Statistics, Lect. Notes Stat., 53, Springer-Verlag, New York (1989).

  2. N. Balakrishnan and A. C. Cohen, Order Statistics and Inference, Academic Press, New York (1991).

    MATH  Google Scholar 

  3. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed., Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester (2003).

  4. Y. Gordon, A. E. Litvak, S. Mendelson, and A. Pajor, “Gaussian averages of interpolated bodies and applications to approximate reconstruction,” J. Approx. Theory, 149, 59–73 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. Gordon, A. E. Litvak, C. Sch¨utt, and E. Werner, “Geometry of spaces between zonoids and polytopes,” Bull. Sci. Math., 126, 733–762 (2002).

    Article  MathSciNet  Google Scholar 

  6. Y. Gordon, A. E. Litvak, C. Sch¨utt, and E. Werner, “Minima of sequences of Gaussian random variables,” C. R. Acad. Sci. Paris, Sér. I Math., 340, 445–448 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Gordon, A. E. Litvak, C. Schütt, and E. Werner, “On the minimum of several random variables,” Proc. Amer. Math. Soc., 134, 3665–3675 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Gordon, A. E. Litvak, C. Schütt, and E. Werner, “Uniform estimates for order statistics and Orlicz functions,” Positivity, 16, 1–28 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. W. Hoeffding, “On the distribution of the number of successes in independent trials,” Ann. Math. Statist., 27, 713–721 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. E. Litvak and K. Tikhomirov, “Order statistics of vectors with dependent coordinates and the Karhunen–Loeve basis,” Ann. Appl. Prob., 28, 2083–2104 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Mallat and O. Zeitouni, “A conjecture concerning optimality of the Karhunen–Loéve basis in nonlinear reconstruction,” arXiv:1109.0489.

  12. P. K. Sen, “A note on order statistics for heterogeneous distributions,” Ann. Math. Statist., 41, 2137–2139 (1970).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Alberta, Edmonton, Canada

    A. E. Litvak

  2. Princeton University, Princeton, USA

    K. Tikhomirov

Authors
  1. A. E. Litvak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Tikhomirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. E. Litvak.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 457, 2017, pp. 265–275.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Litvak, A.E., Tikhomirov, K. Estimates for Order Statistics in Terms of Quantiles. J Math Sci 238, 523–529 (2019). https://doi.org/10.1007/s10958-019-04254-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04254-5

Search

Navigation